,. ' 122 TRAJECTORIES. TSl.

f ,j ;

!/ and therefore

: de ^d® . n

< • r — j^ -__ -}-1 = o

j ar »jK

I || ' where -R and r, © and 0 (but not their derivatives) are -fclxe

same.

Now

~dr d0 dr '

; eliminating c between this equation and the equation of

curve, we find a relation of the form

For the trajectory

' ' dr ^,2^

the differential equation of the trajectory is therefore

This, when integrated, gives the equation of the systeno. of
curves possessing the required property.

] Ex. 1. Find the orthogonal trajectory of the series of straight lines

We have ~=m.

\ j } • <fa'

*' ] s, and- therefore the differential equation of these lines is

i ; xty_

\ Hence, by our rule, the differential*equation of the system of orthogonal

1 . trajectories is

which on integration gives
a series of concentric circles having for common centre the common poixxi> of
the lines. -„ '